Searching of Periodic Components in the Lake Mina, Minnesota Time Series

Transcription

1 Searching of Periodic Components in the Lake Mina, Minnesota Time Series Karim J. Rahim 1, Jeannine St. Jacques 2, David J. Thomson 1 1. Department of Math and Statistics, Queens University 2. Tree Ring Lab, University of Regina TIES Conference 2008 June 10, /33

2 Lake Mina Lake Mina, Minnesota is located just south of the coniferous/deciduous forest ecotone 2/33

3 Lake Mina Diatoms Diatoms are (generally) unicellular microscopic algae with cell walls made of silica. 3/33

4 An example of a Varved Lake Core. Varved Core 4/33

5 The core taken from Lake Mina Lake Mina Core 5/33

6 What are we looking for? We are looking for periodic components in the historical Diatom and Pollen grain record that can be explained by outside forcing. Specifically we are looking for evidence of solar and lunar forcing For example 11-year solar cycle 18.6-year lunar cycle 35-year Brückner solar cycle [3] Evidence of the 52-month ENSO cycle. 6/33

7 We will use the multitaper spectrum estimates[2] Multitaper Multitaper spectral estimates, in there crude form, are averaged direct spectral estimators using Discrete Prolate Spheroidal Sequences (dpss s) of different orders as windowing functions. For each taper one computes the eigencoefficients, y k (f ) = N 1 t=0 The tapers are normalized such that t=0 x(t)v (k) t (N, W)exp( i2πft). (1) N 1 [v (k) t (N, W)] 2 = 1 (2) 7/33

8 Multitaper DPSS Figure 1: DPSS 8/33

9 Multitaper In practice the crude multitaper estimate is not used, instead weights are used to replace averaging the eigencoefficients (1). The weights are calculated iteratively from λk S(f ) d k (f ) λ k S(f ) + B k (f ), (3) where B k (f ) is an estimate of the power in the broad-band bias terms. The initial estimate and bound for the bias term is given by B k (f ) σ 2 (1 λ k ). (4) We also take the multitaper over sectioned overlapped blocks of data and combind these blocks using the arithmetic mean 9/33

10 F-test The multitaper method allows for F-tests test for periodic components in coloured noise. The harmonic F-test is defined, F(f ) = (K 1) ˆµ 2 K 1 k=0 U k(n, W;0) 2 K 1 k=0 y (5) k(f ) ˆµ(f )U k (N, W;0) 2, with 2, and 2K 2 degrees of freedom, where our mean, ˆµ(f ) is estimated by ˆµ(f ) = K 1 k=0 U k(n, W;0)y k (f ) K 1 k=0 U2 k (N, W;0). (6) 10/33

11 Singular Values We take the singular value decomposition of a matrices formed by the eigencoefficients, (1), at each frequency, of most prevalent diatoms by the pollen or diatom samples. Y(f ) is k p Y(f ) = U(f )Σ(f )V (f ), (7) In the general the right singular vectors of 1/ n 1X are the eigenvectors of the covariance matrix X T X, where X is a n p data matrix The multitaper eigencoefficients were obtained using a time-bandwidth parameter of 3.5, and 9 data tapers 11/33

12 Singular Values σ 2 1e 03 1e 01 1e+01 1e+03 Diatoms Period in Years Frequency (cycles/year) σ 2 1e 03 1e 01 1e+01 Pollen Period in Years Frequency (cycles/year) Figure 2: SVD of eigencoefficients and Data 12/33

13 Left Singular Vectors We look at the left singular vectors from the decomposition. We conduct a harmonic F-test, (5) of these singular vectors The left singular vectors are eigenvectors of the outer product matrix M(f ) = Y(f )Y(f ) Each column of M, contains m 1 (f ) = p i=1 y i (f )y (0) i (f ), (8) The left singular vectors with the largest singular values correspond to dimensions in the data with the most variation at a particular frequency 13/33

14 Left Singular Vector Diatom Eigenvector 1 Pollen Eigenvector Cycles/Year Cycles/Year Figure 3: Ftest of Left Eigenvector 1 14/33

15 Left Singular Vector Diatom Eigenvector 2 Pollen Eigenvector Cycles/Year Cycles/Year Figure 4: Ftest of Left Eigenvector 2 15/33

16 Significant Peaks 99.5% SV Data Significant Periods Accountable 1 Pollen Diatom Pollen Diatom Pollen Diatom Pollen Diatom Table 1: Location of Significant Peaks 16/33

17 Canonical Coherence Canonical coherence, which is similar to canonical correlation, using multitaper techniques are relative new They show the linear relationship between two multivariate sets of time series We form the canonical coherence, by taking the SVD of the cross product of left eigenvectors from the original SVD described above 17/33

18 Canonical Coherence Period in Years Coherence (percent) Frequency (cycles/year) Figure 5: Canonical Coherence 18/33

19 Individual Taxa Quercus comprises over one quarter of the pollen sample It is a fire-sensitive deciduous tree taxa that is found both dryer mid-holocene pollen assemblage and colder and moister assemblages [4] Fragilaria Crotonensis comprises 35% of the diatom Sample This Diatom taxa generally peaks twice a year, once in the late spring and a second time in the Autumn The time-bandwidth parameter was set to 3.5, and 7 tapers were used in the Bock Multitaper There are 11 blocks and about 84% overlap 19/33

20 Quercus Period in Years counts 2 (cycles/year) Max Mean Geometric Mean Min Calender Years (a) Quercus Time Series Frequency (cycles/year) (b) Quercus Spectrum Figure 6: Figure 6(a) shows initial the Quercus time series, and figure 6(b) plots the block multitaper spectrum. 20/33

21 Fragilaria Crotonensis Period in Years counts 2 (cycles/year) Calender Years Frequency (cycles/year) (a) Fragilaria Crotonensis Time Series (b) Fragilaria Crotonensis Spectrum Figure 7: Figure 7(a) shows initial the Fragilaria Crotonensis time series, and figure 7(b) plots the block multitaper spectrum. 21/33

22 Quercus Spectrogram db year (cycles/year) Figure 8: Multitaper Spectrogram of Quercus 22/33

23 Fragilaria Crotonensis Spectrogram db year (cycles/year) Figure 9: Multitaper Spectrogram of Fragilaria Crotonensis 23/33

24 Block Values for F-test (cycles/year) (cycles/year) (cycles/year) (cycles/year) (cycles/year) (cycles/year) (cycles/year) (cycles/year) (cycles/year) (cycles/year) Figure 10: Quercus F-tests for each time block 24/33

25 Moving Values for F-test 11.6 Years 11.1 Years 10.9 Years F test F test F test Centred At Year Centred At Year Centred At Year Figure 11: Quercus F-test for fixed frequency 25/33

26 Moving Values for F-test 11.6 Years 11.1 Years 10.9 Years Degrees Centred At Year Degrees Centred At Year Degrees Centred At Year Figure 12: Quercus F-test phase for fixed frequency 26/33

27 Comparing with a Reference Series We can compare coherence with a reference set as a way to help rule out aliases A reference series of annual tree ring growth from Campito Mountain was obtained [1]. We needed four-year resolution so we filtered the reference series by projecting the data onto the spaced spanned by the orthonormal dpss s We form the projection filter x = VV T x. (9) 27/33

28 Filter Using Expansion in dpss s Period in Years (annual growth) 2 (cycles/year) 2 1e 07 1e 04 1e 01 1e+02 1e Frequency (cycles/year) Figure 13: Filtered Campito 28/33

29 Coherence with Reference Series Period in Years Magnitude Squared Coherence Inverse Transform of Magnitude Squared Coherence Cumulative Distribution Function for Independent Data Frequency Figure 14: Coherence with Campito Reference Series 29/33

30 Concluding Remarks We have found several period line components in the data We see evidence of the 11-year and 35-year solar cycle in the two most prevalent of the diatoms and the pollen grains In Quercus we see evidence of an 11=year and 35-year period We also see evidence of a 16-year period which is not as easily explained There are several highly significant period components in the data 30/33

31 Future Work Check for evidence of counting errors Fit a distribution to the possibility of counting errors and test Check for coherence with other reference sets. Methods to present, pictorially, information/results of analysis from multiple species. 31/33

32 References V.C. Lamarche, D.A. Graybill, H.C. Fritts, and M.R. Rose. Increasing Atmospheric Carbon Dioxide: Tree Ring Evidence for Growth Enhancement in Natural Vegetation. Science, 225(4666): , D.B. Percival and A.T. Walden. Spectral analysis for physical applications. Cambridge University Press New York, NY, USA, OM Raspopov, OI Shumilov, EA Kasatkina, E. Turunen, and M. Lindholm. 35-year Climatic Bruckner Cycle-Solar Control of Climate Variability? The solar cycle and terrestrial climate, Solar and space weather, J.M. St. Jacques, B.F. Cumming, and J.P. Smol. A 900-year pollen-inferred temperature and effective moisture record from varved Lake Mina, west-central Minnesota, USA. Quaternary Science Reviews, /33

33 Thanks Thank you 33/33